P-ring

Let $P$ be a commutative ring together with two types of structure on the set-valued functor $Hom(P, -)$ it represents:

- a commutative ring structure, which means we can view it as an endofunctor on the category of commutative rings (so $Spec P$ is a commutative ring scheme), and
- a comonad structure on this endofunctor.

Then a **$P$-ring** is simply a commutative ring equipped with a co-action of this comonad.

This is similar to the situation for operators on abelian groups. Let $A$ be an abelian group; then $Hom(A, -)$ is already abelian-group-valued (since Ab is closed), so we don’t need to do step (1). Then a comonad structure on the endofunctor $Hom(A, -)$ is the same as a (possibly non-commutative) ring structure on $A$.

The moral of the story is that operators on abelian groups naturally form rings, and operators on commutative rings naturally form $P$s. What you call such a $P$ is another matter. Borger and Wieland called such a thing a plethory, with a wink at plethysm, which is another name for composition of symmetric functions. Other people take a historical approach and use ‘Tall–Wraith’ combined with some other words. Probably the most descriptive and generalizable name would be something like ‘composition object in the category of commutative rings’.

Alternatively, the general naming scheme ‘plethystic object? in the category $C$’ could be used, where $C$ could be the category of commutative rings, the category of abelian groups, and so on.

- J. Borger & B. Wieland, Plethystic algebra